Richard S. Sutton, David McAllester, Satinder Singh, Yishay Mansour
Function approximation is essential to reinforcement learning, but the standard approach of approximating a value function and deter(cid:173) mining a policy from it has so far proven theoretically intractable. In this paper we explore an alternative approach in which the policy is explicitly represented by its own function approximator, indepen(cid:173) dent of the value function, and is updated according to the gradient of expected reward with respect to the policy parameters. Williams's REINFORCE method and actor-critic methods are examples of this approach. Our main new result is to show that the gradient can be written in a form suitable for estimation from experience aided by an approximate action-value or advantage function. Using this result, we prove for the first time that a version of policy iteration with arbitrary differentiable function approximation is convergent to a locally optimal policy.
Large applications of reinforcement learning (RL) require the use of generalizing func(cid:173) tion approximators such neural networks, decision-trees, or instance-based methods. The dominant approach for the last decade has been the value-function approach, in which all function approximation effort goes into estimating a value function, with the action-selection policy represented implicitly as the "greedy" policy with respect to the estimated values (e.g., as the policy that selects in each state the action with highest estimated value). The value-function approach has worked well in many appli(cid:173) cations, but has several limitations. First, it is oriented toward finding deterministic policies, whereas the optimal policy is often stochastic, selecting different actions with specific probabilities (e.g., see Singh, Jaakkola, and Jordan, 1994). Second, an arbi(cid:173) trarily small change in the estimated value of an action can cause it to be, or not be, selected. Such discontinuous changes have been identified as a key obstacle to estab(cid:173) lishing convergence assurances for algorithms following the value-function approach (Bertsekas and Tsitsiklis, 1996). For example, Q-Iearning, Sarsa, and dynamic pro(cid:173) gramming methods have all been shown unable to converge to any policy for simple MDPs and simple function approximators (Gordon, 1995, 1996; Baird, 1995; Tsit(cid:173) siklis and van Roy, 1996; Bertsekas and Tsitsiklis, 1996). This can occur even if the best approximation is found at each step before changing the policy, and whether the notion of "best" is in the mean-squared-error sense or the slightly different senses of residual-gradient, temporal-difference, and dynamic-programming methods.
In this paper we explore an alternative approach to function approximation in RL.